Does having lots of add-ons enabled slow down blender?
As mentioned in the comments, the real answer will depend on what addons you enable. Many addons simply add an operator to perform a task and will not add any noticeable difference to the general use of blender, but it is possible for an addon to inject tasks into blenders handlers that will be run almost constantly, if these are slow then it can add a noticeable slow down to many areas of blender.

In general, the more you save as enabled so that they are loaded on startup, the longer blender will take to start, larger addons like animation nodes affect this more than most of the simple addons that are available. Once blender has started and is responding to user input you are unlikely to notice any difference with most addons that are available.

As you tagged your question as cycles related, addons aren't used by cycles and won't affect rendering times. OSL scripts added to a material can affect render times.